Pour rosé and lillet into a 13 x 9" pan and freeze until almost solid (~6h). Scrape rosé into blender. Add juice, syrup, and 1 cup cracked ice. Purée until smooth. Transfer blender jar to freezer and freeze until frosé is thickened (aim for milkshake consistency), 25–35 min. Blend again until frosé is slushy. Divide into glasses and serve with basil sprig.